# Real and imaginary parts of a complex number

I would like to find the real and imaginary parts of $x + y$ in terms of polar coordinates.

Here is code:

ComplexExpand[Re[x + y], {x, y}, TargetFunctions -> {Abs, Arg}]

How can I add the assumption that $Abs[x]$ equals to $Abs[y]$?

For example this code does not work:

ComplexExpand[Re[x + y], {x, y}, TargetFunctions -> {Abs, Arg},
Assumptions -> Abs[x] == Abs[y]]
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By the above assumption, I expect to obtain: Re[x + y] = Abs[x] (Cos[Arg[x]] + Cos[Arg[y]]) –  To Be Jun 21 '14 at 13:23
Or, how can I change the code to obtain Re[x + y] = Abs[x] (Cos[Arg[x]] + Cos[Arg[y]]), by the assumption that Abs[x] = Abs[y]? –  To Be Jun 21 '14 at 13:24